Abstract

We give a characterization of weighted Morrey space by using Fefferman and Stein’s sharp maximal function. For this purpose, we consider a local good- inequality.

1. Introduction

The Fefferman-Stein sharp maximal function (see Definition 2) plays an important role in harmonic analysis. For example, we know , where . We also know , where is the weighted space (for further details, see [1, 2]).

On the other hand, many studies have been done for Morrey spaces: Komori and Shirai [3] introduced weighted Morrey spaces (for the precise definition, see Section 2) and proved the boundedness of the Hardy-Littlewood maximal operator and singular integral operators.

Di Fazio and Ragusa [4] proved From this inequality they obtained However, the condition is strong; that is, the left side of inequality (2) is assumed to be finite. In applications (3) is more important than (2). In this paper, we will prove (3) under weaker condition without using (2). Our proof is different from the one in [4]. We use another method, that is, local good- estimate. We think this itself is interesting (see Section 4, Lemma 16), and we also consider weighted estimates.

This paper is organized as follows. In Section 2 we make some definitions for maximal functions, functions spaces, and weights. In Section 3 we state known results and our theorem. In Section 4 we prove our theorems.

2. Preliminaries

The following notation is used: for a set , we denote the Lebesgue measure of by , and for a nonnegative locally integrable function , we write . We denote the characteristic function of by . Throughout this paper, all cubes are assumed to have their sides parallel to the coordinate axes.

2.1. Maximal Functions

First we define some maximal functions.

Definition 1 (the Hardy-Littlewood maximal function). Considerwhere the supremum is taken over all cubes containing .

Definition 2 (the sharp maximal function). Considerwhere .Next we define the dyadic maximal function. A dyadic cube is a cube of the form

Definition 3 (the dyadic maximal function). Considerwhere the supremum is taken over all dyadic cubes containing .

By (29) and (31), we have Taking sufficiently small, we obtain Since ., this proves the theorem.

5. Proofs of Lemmas

Proof of Lemma 16. Let . As in [2, page 148], for any , there is a maximal dyadic cube that contains such that Since , the maximal dyadic cube satisfies . Therefore, we can write where is a family of disjoint maximal dyadic cubes. We can show By -reverse doubling condition (Definition 8), we obtain the desired result.

Proof of Lemma 19. For any dyadic cubes , and tends to infinity when because (see Proposition 9).

Proof of Lemma 20. Since we haveby reverse doubling condition (Definition 7).By (39), is a Cauchy sequence; therefore, converges to some constant when . If , then . But this contradicts the fact by Lemma 19. Therefore, we obtain .

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.